MathDB

2

Part of 2016 IOM

Problems(1)

International olympiad of metropolises 2016 P2

Source: International olympiad of metropolises 2016

9/7/2016
Let a1,...,ana_1, . . . , a_n be positive integers satisfying the inequality ∑i=1n1an≤12\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}. Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each i=1,...,ni = 1, . . . , n,the possible values of the i−thi-th indicator are 1,2,...,ai1, 2, . . . , a_i. The Annual Report is said to be optimistic if at least n−1n - 1 indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
inequalitiesalgebracombinatorics